Arm exponent for the Gaussian free field on metric graphs in intermediate dimensions

We investigate the bond percolation model on transient weighted graphs $G$ induced by the excursion sets of the Gaussian free field on the corresponding metric graph. We assume that balls in $G$ have polynomial volume growth with growth exponent $\alpha$ and that the Green’s function for the random on $G$ exhibits a power law decay with exponent $\nu$, in the regime $1\le \nu \le \frac{\alpha }{2}$. In particular, this includes the cases of $G={\mathbb{Z}}^3$ for which $\nu =1$, and $G={\mathbb{Z}}^4$ for which $\nu =\frac{\alpha }{2}=2$. For all such graphs, we determine the leading-order asymptotic behavior for the critical one-arm probability, which we prove decays with distance $R$, like $R^{-\frac{\nu }{2}+o(1)}$. Our results are, in fact, more precise and yield logarithmic corrections when $\nu >1$ as well as corrections of order $\log \log R$ when $\nu =1$. We further obtain very sharp upper bounds on truncated two-point functions close to criticality, which are new when $\nu >1$ and essentially optimal when $\nu =1$. This extends previous results from arXiv:2101.05801 and arXiv:1807.11117.